MAT224H1c.doc

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Inner Product Spaces

INNER PRODUCTS

Definition

An inner product on a vector space V is a function that assigns a number wv, to every pair v, w of vector

space V in such a way that the following axioms holds:

P1: wv, is a real number.

P2: vwwv,, =.

P3: uwuvuwv,,, +=+ .

P4: wvrwrv,, =.

P5: Vvvv ∈∀≥,0,.

Definition

A vector space V with an inner product is called an inner product space.

Note

• R→×VV:,.

•

(

)

⋅+,,, RV is a vector space.

•

(

)

,,,,, ⋅+RV is an inner product space.

Examples

The following are inner product spaces.

1)

(

)

,,,,, ⋅+RR n, define YXYX⋅=, the dot product.

2)

[

]

(

)

,,,,,, ⋅+RbaC, define

( ) ( )

=b

adxxgxfgf,.

3)

(

)

,,,,, ⋅+

×R

mn

M, define

(

)

T

ABBA tr,=.

Theorem

Let , be an inner product on a space V. Let u, v, w denote vectors in V, r a real number. Then:

1) wuvuwvu,,, +=+ .

2) wvrrwv,, =.

3) vv ,000,== .

4) 0,=vv if and only if 0

=

v.

Theorem

If A is any n×n positive definite matrix, then nTYXAYXYXR∈∀=,,, defines an inner product on Rn,

and every inner product on Rn, and every inner product on Rn arises in this way.

Proof:

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MAT224H1c.doc

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• AYXYXT

=, is an inner product.

• Any , on Rn can be expressed as

AY

X

T:

• Let

{

}

n

EEE ,,

1= be the standard basis of Rn. Then

=

=

n

i

ii ExX

1

and

=

=

n

j

jj EyY

1

.

•

[ ]

AYX

y

y

EEEE

EEEE

xx

EEyxEyExYX

T

nnnn

n

n

n

ji

jiji

j

j

jj

n

i

ii

=

=

==

===

1

1

111

1

1,11

,,

,,

,,,

• Moreover, T

A

A

=

.

NORMS AND DISTANCE

Definition

The norm of v in V is defined as

( )

vvvv ,norm== (length).

Definition

The distance between vectors v, w in an inner product space is

(

)

wvwv−=,d.

Theorem

If 0

≠

v is any vector in an inner product space, then v

v

v=

ˆ is the unique unit vector that is a positive

multiple of V.

Theorem: Schwarz Inequality

If v and w are two vectors in an inner product space V, then 222

,wvwv≤. Moreover, equality occurs if

and only if one of v or w is a scalar multiple of the other.

Proof:

• Assume 0>= av and 0>= bw.

•

(

)

abwvwvababawbvawbvawbv≤≥−=−−=− ,0,2,

2, and

(

)

abwvwvababawbvawbvawbv−≥≥+=++=+ ,0,2,

2.

• So 22

2

2

,0,wvbawvabwvab b=≤≤≤≤− .

• Note: 1

,

22

2

≤

wv

wv or 1

,

1≤≤− wv

wv.

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Example

Consider the vector space

[

]

baC, of all continuous functions on

[

]

ba, . Define

( ) ( )

=b

adxxgxfgf,.

Then

( ) ( ) ( )( ) ( )( )

⋅≤

b

a

b

a

b

adxxgdxxfdxxgxf22

2

.

Theorem: Properties of Norms

1) 0≥v.

2) 0=v if an only if 0

=

v.

3) vrrv=.

4) wvwv+≤+ (triangle inequality).

Theorem: Properties of Distance

1)

(

)

0,d≥wv.

2)

(

)

0,d=wv if and only if

w

v

=

.

3)

(

)

(

)

vwwv,d,d=.

4)

(

)

(

)

(

)

Vwuvwuuvwv∈∀+≤ ,,,,d,d,d

ORTHOGONAL SETS OF VECTORS

Definition

Two vectors v, w in an inner product space V are said to be orthogonal if 0,=wv.

Definition

A set of vectors

{

}

n

ee ,,

1 is called an orthogonal set if each 0≠

i

e and jiee ji ≠∀=,0,. If, in

addition, iei∀=,1, then the set is called an orthonormal set.

Example

Consider

{

}

xx cos,sin in

[

]

ππ

,−C with

( ) ( )

−

=

π

π

dxxgxfgf,. Then 0cos,sin=xx , so

{

}

xx cos,sin is an orthogonal set.

Theorem: Pythagorean Theorem

If

{

}

n

ee ,,

1 is an orthogonal set of vectors, then 22

1

2

1nn eeee ++=++ .

Theorem

Let

{

}

n

ee ,,

1 be an orthogonal set of vectors. Then:

1)

{

}

nn erer,,

11 is also orthogonal for all 0≠

i

r in R.

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